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Mirrors > Home > MPE Home > Th. List > vccl | Structured version Visualization version GIF version |
Description: Closure of the scalar product of a complex vector space. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
vciOLD.1 | ⊢ 𝐺 = (1st ‘𝑊) |
vciOLD.2 | ⊢ 𝑆 = (2nd ‘𝑊) |
vciOLD.3 | ⊢ 𝑋 = ran 𝐺 |
Ref | Expression |
---|---|
vccl | ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋) → (𝐴𝑆𝐵) ∈ 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vciOLD.1 | . . 3 ⊢ 𝐺 = (1st ‘𝑊) | |
2 | vciOLD.2 | . . 3 ⊢ 𝑆 = (2nd ‘𝑊) | |
3 | vciOLD.3 | . . 3 ⊢ 𝑋 = ran 𝐺 | |
4 | 1, 2, 3 | vcsm 28911 | . 2 ⊢ (𝑊 ∈ CVecOLD → 𝑆:(ℂ × 𝑋)⟶𝑋) |
5 | fovrn 7434 | . 2 ⊢ ((𝑆:(ℂ × 𝑋)⟶𝑋 ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋) → (𝐴𝑆𝐵) ∈ 𝑋) | |
6 | 4, 5 | syl3an1 1162 | 1 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋) → (𝐴𝑆𝐵) ∈ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 × cxp 5584 ran crn 5587 ⟶wf 6424 ‘cfv 6428 (class class class)co 7269 1st c1st 7820 2nd c2nd 7821 ℂcc 10858 CVecOLDcvc 28907 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 ax-un 7580 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3433 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-nul 4259 df-if 4462 df-sn 4564 df-pr 4566 df-op 4570 df-uni 4842 df-br 5076 df-opab 5138 df-mpt 5159 df-id 5486 df-xp 5592 df-rel 5593 df-cnv 5594 df-co 5595 df-dm 5596 df-rn 5597 df-iota 6386 df-fun 6430 df-fn 6431 df-f 6432 df-fv 6436 df-ov 7272 df-1st 7822 df-2nd 7823 df-vc 28908 |
This theorem is referenced by: vc0 28923 vcm 28925 nvscl 28975 |
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